Transactions of the American Mathematical Society, 1976
The possibility (subject to certain restrictions) of solving the following approximation and inte... more The possibility (subject to certain restrictions) of solving the following approximation and interpolation problem with a given set of “Muntz polynomials” on a real interval is demonstrated: (i) approximation of a continuous function by a “copositive” Muntz polynomial; (ii) approximation of a continuous function by a “comonotone” Muntz polynomial; (iii) approximation of a continuous function with a monotone kth difference by a Muntz polynomial with a monotone kth derivative; (iv) interpolation by piecewise monotone Muntz polynomials—i. e., polynomials that are monotone on each of the intervals determined by the points of interpolation. The strong interrelationship of these problems is shown implicitly in the proofs.
For a set of monotone (and/or convex) data, we consider the possibility of finding a spline inter... more For a set of monotone (and/or convex) data, we consider the possibility of finding a spline interpolant, of pre-determined smoothness, which is monotone (and/or convex). The investigation is carried out by constructing an auxiliary set of points and using the well-known monotonicity and convexity preserving properties of Bernstein polynomials. In § 3 we consider the problem of piecewise monotone interpolation.
Let 0 =< < < < be a sequence of integers. Necessary and sufficient conditions are obtained for {x... more Let 0 =< < < < be a sequence of integers. Necessary and sufficient conditions are obtained for {xt, xt', xt"} to form an extended Chebyshev system of order n + on (- ,).
Let $F = \{ {f_i } \}_{i = 0}^n $ be a set of continuous functions on $[a,b]$, and let $F^ * = \{... more Let $F = \{ {f_i } \}_{i = 0}^n $ be a set of continuous functions on $[a,b]$, and let $F^ * = \{ {f_i f_j } \}_{i,j = 0}^n $. We determine conditions on F which are necessary and sufficient for the set $F^ * $ to be a Chebyshev system on $[a,b]$ consisting of exactly $2n + 1$ distinct functions. The results have applications in the field of experimental design.
Proceedings of the American Mathematical Society, 1970
We show that on the curve y = x α , α y = {x^\alpha },\alpha any irrational, 0 ≦ x ≦ 1 0 \leqq x ... more We show that on the curve y = x α , α y = {x^\alpha },\alpha any irrational, 0 ≦ x ≦ 1 0 \leqq x \leqq 1 , the degree of approximation by n n th degree polynomials in x x and y y in the L 2 {L^2} norm has order of magnitude 1 / n 3 / 2 1/{n^{3/2}} .
Proceedings of the American Mathematical Society, 1974
Jackson type theorems are obtained for monotone and comonotone approximation. Namely (i) If/(;r) ... more Jackson type theorems are obtained for monotone and comonotone approximation. Namely (i) If/(;r) is a function such that the kth difference of / is =ï0 on [a, b] then the degree of approximation of/by nth degree polynomials with kth derivative ^0 is 0[a>(/; l/«1-*)] for any e>0, where a>(f; <5) is the modulus of continuity oí fon [a, b]. (ii) If f(x) is piecewise monotone on [a, b) then the degree of approximation of/ by nth degree polynomials comonotone with / is 0[o(/; I/«1"*)] for any e>0.
Algorithms are presented for computing a smooth piecewise polynomial interpolation which preserve... more Algorithms are presented for computing a smooth piecewise polynomial interpolation which preserves the monotonicity and/or convexity of the data.
Let P, be the set of all algebraic polynomials of degree II or less. For f~ C[u, b], the degree o... more Let P, be the set of all algebraic polynomials of degree II or less. For f~ C[u, b], the degree of approximation to f by polynomials in P, is En(f) = inf{llf-p 11: p E PJ, where the norm is the uniform norm. Jackson's theorem [l] states that there exists C > 0 such that En(f) < Cw(f; l/n), where w(f; 6) is the modulus of continuity off. f is said to be piecewise monotone if it has only a finite number of local maxima and minima in [a, b]. The local maxima and minima in (a, b) are called the peaks off. Wolibner [5] has shown that for any E > 0 there exists a polynomial, p, such that IIfp II < E and p is comonotone with f; i.e., p increases and decreases simultaneously with $ Let E,*(f) = inf{llf-p 11: p E P, , p comonotone with f}. Clearly E,*cf) 3 E,df). We seek an upper bound on E,*(f). Forfmonotone, Lorentz and Zeller [3] have shown that there exists C, > 0 such that E,*(f) < C,w(f; I/n). Newman, Passow, and Raymon [4] have obtained results of a modified nature. They have shown that there exists p E P, satisfying IIfp II < C+(f; l/n), C, an absolute constant, such that f and p are comonotone except in certain neighborhoods (whose diameters tend to zero with n) of the peaks. In this note we obtain a comonotone approximation on the entire interval [a, b], but at a sacrifice in the accuracy of approximation. LEMMA 1. Let fe C(j+l)[a, b] and suppose that f(u) = 0, u E (a, b). Let g(x) =f[x, u], the divided dzfirence off, where we dejine f[u, U] =f'(u). Then g E Cj[u, b] and II g(j) 11 < (j + 1)-l /lf(j+') 11.
Transactions of the American Mathematical Society, 1976
The possibility (subject to certain restrictions) of solving the following approximation and inte... more The possibility (subject to certain restrictions) of solving the following approximation and interpolation problem with a given set of “Muntz polynomials” on a real interval is demonstrated: (i) approximation of a continuous function by a “copositive” Muntz polynomial; (ii) approximation of a continuous function by a “comonotone” Muntz polynomial; (iii) approximation of a continuous function with a monotone kth difference by a Muntz polynomial with a monotone kth derivative; (iv) interpolation by piecewise monotone Muntz polynomials—i. e., polynomials that are monotone on each of the intervals determined by the points of interpolation. The strong interrelationship of these problems is shown implicitly in the proofs.
For a set of monotone (and/or convex) data, we consider the possibility of finding a spline inter... more For a set of monotone (and/or convex) data, we consider the possibility of finding a spline interpolant, of pre-determined smoothness, which is monotone (and/or convex). The investigation is carried out by constructing an auxiliary set of points and using the well-known monotonicity and convexity preserving properties of Bernstein polynomials. In § 3 we consider the problem of piecewise monotone interpolation.
Let 0 =< < < < be a sequence of integers. Necessary and sufficient conditions are obtained for {x... more Let 0 =< < < < be a sequence of integers. Necessary and sufficient conditions are obtained for {xt, xt', xt"} to form an extended Chebyshev system of order n + on (- ,).
Let $F = \{ {f_i } \}_{i = 0}^n $ be a set of continuous functions on $[a,b]$, and let $F^ * = \{... more Let $F = \{ {f_i } \}_{i = 0}^n $ be a set of continuous functions on $[a,b]$, and let $F^ * = \{ {f_i f_j } \}_{i,j = 0}^n $. We determine conditions on F which are necessary and sufficient for the set $F^ * $ to be a Chebyshev system on $[a,b]$ consisting of exactly $2n + 1$ distinct functions. The results have applications in the field of experimental design.
Proceedings of the American Mathematical Society, 1970
We show that on the curve y = x α , α y = {x^\alpha },\alpha any irrational, 0 ≦ x ≦ 1 0 \leqq x ... more We show that on the curve y = x α , α y = {x^\alpha },\alpha any irrational, 0 ≦ x ≦ 1 0 \leqq x \leqq 1 , the degree of approximation by n n th degree polynomials in x x and y y in the L 2 {L^2} norm has order of magnitude 1 / n 3 / 2 1/{n^{3/2}} .
Proceedings of the American Mathematical Society, 1974
Jackson type theorems are obtained for monotone and comonotone approximation. Namely (i) If/(;r) ... more Jackson type theorems are obtained for monotone and comonotone approximation. Namely (i) If/(;r) is a function such that the kth difference of / is =ï0 on [a, b] then the degree of approximation of/by nth degree polynomials with kth derivative ^0 is 0[a>(/; l/«1-*)] for any e>0, where a>(f; <5) is the modulus of continuity oí fon [a, b]. (ii) If f(x) is piecewise monotone on [a, b) then the degree of approximation of/ by nth degree polynomials comonotone with / is 0[o(/; I/«1"*)] for any e>0.
Algorithms are presented for computing a smooth piecewise polynomial interpolation which preserve... more Algorithms are presented for computing a smooth piecewise polynomial interpolation which preserves the monotonicity and/or convexity of the data.
Let P, be the set of all algebraic polynomials of degree II or less. For f~ C[u, b], the degree o... more Let P, be the set of all algebraic polynomials of degree II or less. For f~ C[u, b], the degree of approximation to f by polynomials in P, is En(f) = inf{llf-p 11: p E PJ, where the norm is the uniform norm. Jackson's theorem [l] states that there exists C > 0 such that En(f) < Cw(f; l/n), where w(f; 6) is the modulus of continuity off. f is said to be piecewise monotone if it has only a finite number of local maxima and minima in [a, b]. The local maxima and minima in (a, b) are called the peaks off. Wolibner [5] has shown that for any E > 0 there exists a polynomial, p, such that IIfp II < E and p is comonotone with f; i.e., p increases and decreases simultaneously with $ Let E,*(f) = inf{llf-p 11: p E P, , p comonotone with f}. Clearly E,*cf) 3 E,df). We seek an upper bound on E,*(f). Forfmonotone, Lorentz and Zeller [3] have shown that there exists C, > 0 such that E,*(f) < C,w(f; I/n). Newman, Passow, and Raymon [4] have obtained results of a modified nature. They have shown that there exists p E P, satisfying IIfp II < C+(f; l/n), C, an absolute constant, such that f and p are comonotone except in certain neighborhoods (whose diameters tend to zero with n) of the peaks. In this note we obtain a comonotone approximation on the entire interval [a, b], but at a sacrifice in the accuracy of approximation. LEMMA 1. Let fe C(j+l)[a, b] and suppose that f(u) = 0, u E (a, b). Let g(x) =f[x, u], the divided dzfirence off, where we dejine f[u, U] =f'(u). Then g E Cj[u, b] and II g(j) 11 < (j + 1)-l /lf(j+') 11.
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